Diagram of the extended surfaces denoted by 1 and 2: (a) before contact and (b) after deformation. The positions of their mean surfaces before and after deformation are shown with dashed lines.

Schematic description of asperities in contact (a) and the corresponding contact force distribution (b). Dotted lines represent the undeformed shapes of the asperities. Solid lines represent their deformed slopes. The positions of the matching surface points before the contact and after the deformation are denoted with 1 and 2, and 1* and 2* , respectively, where δ−h(x,y) describes their contact approach.

Flow chart of the computer program that calculates iteratively contact parameters between two extended surfaces using the three-stage loop to solve for self-consistent displacement and pressure distributions

Substructuring of the contact domain between nominally flat surface and wavy and rough surface based on the K-D notion of surface contacts 18: the distribution of contour areas of contact over the nominal contact area of size 1024 mm2 (a), the distribution of asperity contacts inside a contour area of contact (b), and an asperity contact (c). The external normal load P is equal to 300 N. The asperities of the equivalent rough surface have the standard deviation σe=0.43 μm. The surface waviness has wavelength λw(c)=11 mm and wave amplitude hw(c)=6 μm. The sizes of traction elements used for the computations are as follows: dx(1)=dy(1)=62.5 μm,dx(2)=dy(2)=6.25 μm, and dx(3)=dy(3)=0.2 μm.

Predicted variation of true contact area between two extended surfaces (A0=1024 mm2 and E′=83 GPa) with external load. Solid lines represent the results from the present method, where symbols ○ and □ mark the results for the first set (where surfaces have λw(s)=17 mm,hw(s)=8 μm, and σ(s)=0.35 μm and λw(c)=11 mm,hw(c)=6 μm, and σ(c)=0.25 μm) and the second set (where surfaces have σ(s)=0.35 μm and λw(c)=11 mm,hw(c)=6 μm, and σ(c)=0.25 μm) of surface topographies, respectively. Dotted line—the variation of true contact area given by the contact model presented in 17; dashed line—the G-W model 6, and dashed dotted line—the model by Bowden and Tabor 1.

Variation of contact parameters between extended surfaces (A0=1024 mm2 and E′=83 GPa) with load predicted by the present method for two sets of surface topographies: (a) number of asperity contacts, Γ, where Γ=∑jkmj; (b) mean contact area per asperity contact normalized with respect to the mean area of the asperity base, πRe2, where Re=0.99 mm; (c) mean contact pressure normalized with respect to the hardness of the softer solid (Hs=800 MPa), and (D) mean distance between asperity contacts normalized with respect to the correlation radius ρ (ρ=45 μm). The results for the first set of topographies (where surfaces have λw(s)=17 mm,hw(s)=8 μm, and σ(s)=0.35 μm and λw(c)=11 mm,hw(c)=6 μm, and σ(c)=0.25 μm) are marked with ○’s while the corresponding results for the second set (where surfaces have σ(s)=0.35 μm and λw(c)=11 mm,hw(c)=6 μm, and σ(c)=0.25 μm) are marked with □’s. The dashed lines show the corresponding results from the G-W model 6, for comparison.

Contact approach between two extended surfaces (E′=83 GPa) versus external load, where symbols ○ and □ mark the results for the first set (where surfaces have λw(s)=17 mm, hw(s)=8 μm, and σ(s)=0.35 μm and λw(c)=11 mm, hw(c)=6 μm, and σ(c)=0.25 μm) and the second set (where surfaces have σ(s)=0.35 μm and λw(c)=11 mm, hw(c)=6 μm, and σ(c)=0.25 μm) of surface topographies, respectively.

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